Exponential growth of homotopy groups of suspended finite complexes

Abstract

We study the asymptotic behavior of the homotopy groups of simply connected finite p-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the functorial decomposition of loop suspension, we prove that the suspended finite complexes are locally hyperbolic if suitable but accessible information of the homotopy groups is known. In particular, we prove that Moore spaces are locally hyperbolic, and other candidates are also given.2010 Mathematics Subject Classification. primary 55Q52; secondary 55Q20, 55P35, 55P40, 55Q15, 55T15.

p-Hyperbolicity of homotopy groups via K-theory

“…This criterion is quite different to that given by Huang and Wu [20,Theorem 1.5]. Their criterion is homotopical, using hypotheses on X to produce retracts of X , whereas ours is cohomological, which makes it easier to check.…”

“…A space X is called rationally elliptic if π * (X ) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m π i (X ) ⊗ Q grows exponentially in m. It was proved in [12,Chapter 33] that simply connected CW -complexes with rational homology of finite type and finite rational category are either rationally elliptic or rationally hyperbolic. In order to study the p-torsion analogue of this dichotomy, Huang and Wu [20] introduced the definitions of Z/ p r -and p-hyperbolicity.…”

“…Acknowledgements I would like to thank my PhD supervisor, Stephen Theriault, for suggesting the problems that this paper tries to address, and for many helpful conversations along the way. From a technical point of view, much is owed to papers of Huang and Wu [20], and of Selick [28]. Neil Strickland's 'Bestiary' was extremely helpful in providing examples of Theorem 2.…”

$p$-hyperbolicity of homotopy groups via $K$-theory

“…Let X be a path connected space having the homotopy type of a finite CW -complex, and let p be an odd prime. Suppose that there exists a map µ 1 ∨ µ 2 : S q1+1 ∨ S q2+1 → ΣX with q i ≥ 1, such that the map This criterion is quite different to that given by Huang and Wu [HW,Theorem 1.5]. Their criterion is homotopical, using hypotheses on X to produce retracts of ΩΣX, wheras ours is cohomological, which makes it easier to check.…”

“…Huang and Wu show that for n ≥ 3, r ≥ 1 and p any prime, the Moore space P n (p r ) is Z/p rhyperbolic and Z/p r+1 -hyperbolic, and that P n (2) is also Z/8-hyperbolic [HW,Theorem 1.6]. More generally, they give criteria in terms of a functorial loop space decomposition due to Selick and Wu [SW00; SW06] for a suspension ΣX to be Z/p r -hyperbolic.…”